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Simplifying 4q2 + 8q + -1 = 0 Reorder the terms: -1 + 8q + 4q2 = 0 Solving -1 + 8q + 4q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 2q + q2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 2q + 0.25 + q2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 2q + q2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 2q + q2 = 0 + 0.25 2q + q2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 2q + q2 = 0.25 The q term is 2q. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2q + 1 + q2 = 0.25 + 1 Reorder the terms: 1 + 2q + q2 = 0.25 + 1 Combine like terms: 0.25 + 1 = 1.25 1 + 2q + q2 = 1.25 Factor a perfect square on the left side: (q + 1)(q + 1) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (q + 1) equal to 1.118033989 and -1.118033989.Subproblem 1
q + 1 = 1.118033989 Simplifying q + 1 = 1.118033989 Reorder the terms: 1 + q = 1.118033989 Solving 1 + q = 1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + q = 1.118033989 + -1 Combine like terms: 1 + -1 = 0 0 + q = 1.118033989 + -1 q = 1.118033989 + -1 Combine like terms: 1.118033989 + -1 = 0.118033989 q = 0.118033989 Simplifying q = 0.118033989Subproblem 2
q + 1 = -1.118033989 Simplifying q + 1 = -1.118033989 Reorder the terms: 1 + q = -1.118033989 Solving 1 + q = -1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + q = -1.118033989 + -1 Combine like terms: 1 + -1 = 0 0 + q = -1.118033989 + -1 q = -1.118033989 + -1 Combine like terms: -1.118033989 + -1 = -2.118033989 q = -2.118033989 Simplifying q = -2.118033989Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.118033989, -2.118033989}
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